The present invention relates generally to the field of medical imaging and more particularly to a technique for reconstructing PET scan images.
Positron emission tomography (PET) is a technique commonly used in clinical medicine and biomedical research to create images of a living body in its active state. PET scanners can produce images that illustrate various biological process and functions. In a PET scan, the patient is initially injected with a radiotracer. A radiotracer comprises bio-chemical molecules that are tagged with a positron emitting radioisotope and can participate in certain physiological processes in the patient's body. Typical positron-emitting PET isotopes include 11C, 13N, 15O and 18F. When positrons are emitted within the body, they combine with electrons in the neighboring tissues and annihilate. The annihilation events usually result in pairs of gamma photons, with 511 keV of energy each, being released in opposite directions. The gamma photons are then detected by a detector ring like the one shown in FIG. 1. The detector ring 100 may comprise a number of detectors or detector channels (e.g., 11, 12, 13, and 14 etc.) each having a scintillator block and a photomultiplier tube (PMT). For example, detector 11 comprises a scintillator block 112 and a PMT 114, detector 12 comprises a scintillator block 122 and a PMT 124, and so on. In another configuration, a detector block may comprise a matrix of individual scintillator crystals which are read out by a matrix of PMTs. For example, scintillator block 122 may comprise 36 crystals arranged in a square which are read-out by a 2×2 matrix of PMTs. The matrix of PMTs is able to identify an individual scintilator crystal in which a gamma photon is detected.
In operation, a patient 102, who has been injected with a PET radiotracer, may be positioned in the detector ring 100. One pair of gamma photons from a body part 104 may be detected by 2 detectors 11 and 12. The pair of detectors constitutes a line of response (LOR) 116. Another pair of gamma photons from the body part 104 may be detected along another LOR 136. When detected, each of the gamma photons produces numerous optical photons inside its corresponding scintillation blocks 112 and 122. Along the LOR 116, the gamma photons may cause substantially simultaneous scintillations in the scintillator blocks 112 and 122. These scintillations may then be amplified and converted into electrical signals by the PMTs 114 and 124 respectively. Subsequent electronic circuitry may determine whether these substantially simultaneous scintillations are coincidence events, that is, radiation events originating from the same annihilation event in the patient 102's body. Data associated with coincidence events along a number of LORs may be collected and further processed to reconstruct two-dimensional (2-D) tomographic images. Some modern PET scanners can operate in a three-dimensional (3-D) mode, where coincidence events from different detector rings positioned along the axial direction are counted to obtain 3-D tomographic images. An exemplary PET scanner with multiple detector rings is shown in FIG. 2, where the PMTs are not shown. As shown, the PET scanner 200 comprises three detector rings 22, 24 and 26.
Traditionally, data associated with coincidence events are stored in the form of sinograms based on their corresponding LORs. For example, in a 2-D PET scanner like the one illustrated in FIG. 3, if a pair of coincidence events are detected by two opposite detectors 302 and 304, an LOR may be established as a straight line 306 linking the two detectors. This LOR may be identified by two coordinates (r, θ), wherein r is the radial distance of the LOR from the center axis of the detector ring 300, and θ is the trans-axial angle between the LOR and the X-axis. The detected coincidence events may be recorded in a 2-D matrix λ(r, θ). As the PET scanner continues to detect coincidence events along various LORs, these events may be binned and accumulated in their corresponding elements in the matrix λ(r, θ). The result is a 2-D sinogram λ(r, θ), each element of which holds an event count for a specific LOR. In a 3-D PET scanner, an LOR is defined by four coordinates (r, θ, φ, z), wherein the third coordinate φ is the axial angle between the LOR and the center axis (or Z-axis as shown in FIG. 2) of the detector rings and z is the distance of the LOR from the center of detector along the Z-axis. Typically the third and fourth co-ordinates are combined into only one variable, v, which can define both φ and z coordinates. In this case, the detected coincidence events are stored in a 3-D sinogram λ(r, θ, v).
In addition to the true coincidence events described above, two other types of coincidence events are detected by the PET scanner, and these events confound the data collection and image reconstruction process. The first type of confounding events arises because the annihilation photons may scatter as they travel out of the patient. If one or both of the annihilation photons scatter, and are subsequently detected in coincidence, they will register a coincidence event along an LOR that does not correspond to the site of the annihilation event. These events are called “scattered coincidences.” A scattered coincidence may be differentiated from a true coincidence in that scattered photons have energy less than 511 keV. However, due to practical considerations in the design of PET detectors, the energy of each detected photon cannot be measured exactly, and some scattered photons, and therefore some scattered coincidences, are accepted by the scanner. It therefore becomes necessary in the reconstruction process to estimate the contribution of scattered coincidences to the acquired sinograms. Several well-known methods exist for scattered coincidence estimation for both 2D and 3D sinogram sets.
The second type of confounding coincidence event arises from the essentially simultaneous detection of two photons that arose from two different annihilation events that occur at nearly the same time. These coincidences are called “random coincidences.” The contribution of random coincidences may be reduced by reducing the timing window used to define the simultaneous detection of the coincident photons, but some random coincidences will be accepted by the scanner. The rate of random coincidence acceptance may be estimated by two well-known methods. In one of these methods, known as “Delayed Window Method”, the signal from one of the detectors is delayed in time, and a second coincidence circuit finds coincidences between this signal and the undelayed signal from the second detector. The expected rate of coincidences from this delayed coincidence channel is equal to the expected rate of random coincidences from the undelayed, or “prompt,” channel. In the second method of random coincidence estimation, known as the “Randoms from Singles Method”, the detected event rate for each detector is measured and used to predict the random coincidence rate in the prompt channel.
The sinogram data collected by a PET scanner is not an image that illustrates the spatial distribution of the radiotracer inside the patient. Certain mathematical operations must be performed on the sinogram data to transform them into images. The process of generating images is called “tomographic image reconstruction.” The image reconstruction techniques are based on the fact that, when a coincidence event is detected by two detectors, the emission point must have occurred somewhere along the line (i.e., LOR) joining the two detectors. Various techniques are known in the literature for tomographic image reconstruction. The techniques range from analytical methods which are computationally efficient to iterative reconstruction techniques which can be computationally demanding.
Two standard iterative reconstruction algorithms used in PET image reconstruction are maximum likelihood expectation maximization (MLEM) and ordered subsets expectation maximization (OSEM), which can be described by the following iterative update equations (1) and (2) respectively.
                                          F            ^                    j                      k            +            1                          =                                                            F                ^                            j              k                                                      ∑                i                            ⁢                                                          ⁢                              P                ij                                              ⁢                                    ∑              i                        ⁢                                                  ⁢                                                            λ                  i                                ⁢                                  P                  ij                                                                                                  ∑                                          j                      ′                                                        ⁢                                                                          ⁢                                                            P                                              ij                        ′                                                              ⁢                                                                  F                        ^                                                                    j                        ′                                            k                                                                      +                                  S                  i                                +                                  R                  i                                                                                        (        1        )                                                      F            ^                    j                      k            ,                          m              +              1                                      =                                                            F                ^                            j                              k                ,                m                                                                    ∑                                  i                  ∈                                      s                    m                                                              ⁢                                                          ⁢                              P                ij                                              ⁢                                    ∑                              i                ∈                                  s                  m                                                      ⁢                                                  ⁢                                                            λ                  i                                ⁢                                  P                  ij                                                                                                  ∑                                          j                      ′                                                        ⁢                                                                          ⁢                                                            P                                              ij                        ′                                                              ⁢                                                                  F                        ^                                                                    j                        ′                                            k                                                                      +                                  S                  i                                +                                  R                  i                                                                                        (        2        )            wherein {circumflex over (F)}jk is an estimate of the image (e.g., number of annihilation events) at location j on the kth itheration, λi is the number of prompt coincidence events detected by the ith LOR, Si is the number of estimated scatter events detected by the ith LOR, Ri is the number of estimated random events detected by the ith LOR, Pij is a system matrix that determines the probability that activity from location j is detected by the ith LOR, and Sm is the mth subset of LORs.
These equations represent iteration steps of: (a) transforming an estimate of the reconstructed image into an estimated sinogram, (b) adding the estimate of the scatter and random coincidences, (c) comparing the estimated sinogram with the measured prompts sinogram to generate a correction sinogram, (d) generating a correction to the estimated image by back-projecting the correction sinogram, and (e) updating the estimated image based on the correction.
A well-known improvement to PET detection methods is Time-Of-Flight PET (TOF PET), where, in addition to coincidence detection, the difference in the detection time of the individual photons in the coincidence pair is measured. In TOF PET, upon detection of a radiation event (e.g., a gamma photon), the scintillator block at the detection locale time-stamps the detected radiation event. Since both the photons travel at the speed of light, the difference in their time stamps can be used to better localize the annihilation event along the LOR. In a TOF-PET scanner, the position of the emission event relative to the midpoint between the two detectors is determined fromx=c(t1−t2)/2  (3)where, t1 and t2 are the detection times of the two photons and c is the speed of light. In reality, TOF information can only be measured within a certain uncertainty dictated by the timing resolution of the detectors. Consequently, the emission event can be localized probabilistically to a short line segment. The uncertainty in event localization is given byΔx=cΔt/2  (4)where Δx is the location uncertainty and Δt is the timing resolution. In non-TOF PET, the detection times are ignored and the annihilation is equally probable to have occurred along the full extension of the LOR. Incorporation of the time of flight information helps localize the actual emission point for each event, thereby reducing statistical uncertainty in the reconstructed images.
The sinogram data in TOF PET, have an additional coordinate, Δt, which is the difference in the time stamps of two detected photons. Therefore TOF sinogram can be λ(r, θ, Δt) for a 2-D PET scanner and λ(r, θ, v, Δt) for a 3-D PET scanner. The added dimensions of the sinogram data in 3-D PET or TOF PET can greatly increase computational demands in the tomographic image reconstruction. For example, in one particular 3-D PET scanner at one level of quantization, an LOR may have 249 possible values for the radial distance r, 210 possible values for the trans-axial angle θ, and 553 possible values for the axial angle φ. In TOF mode, with a time-stamp quantization of 250 picoseconds (ps), 21 sinogram bins are required along the t dimension to cover a 70 cm field of view (FOV). As a result, this 3-D TOF PET scanner distinguishes over 600×106 different LORs and its sinogram λ(r, θ, φ, t) contains over 6×108 elements. For another detector geometry, a sinogram can have about 1.4×109 elements. In addition, considering that image reconstruction will require a sinogram each for prompts, scatter and randoms coincidences, this can result in a significant computational and data storage burden. With the development of PET technology, the number of LORs or the size of the sinograms are expected to increase even more.
On the other hand, the PET sinograms usually are sparsely populated. For example, a 300-second data acquisition at an event count rate of 50 kcps (kilo counts per second) will result in only 15 million detected events. For a sinogram with over 6×108 elements, only a small fraction of the sinogram elements are non-zero. However, during image reconstruction, computations are performed on all the sinogram elements, whether zero or non-zero, resulting in a considerable waste of computation resources on empty sinogram bins.
List-mode image reconstruction has been proposed as a computationally efficient and data storage friendly solution for conventional (non-TOF) scanners. See, e.g., Lucas Parra, Harrison Barrett, “List-Mode Likelihood: EM Algorithm and Image Quality Estimation Demonstrated on 2-D PET”, IEEE Transaction in Medical Imaging, Vol. 17, No. 2, pp. 228-235, April 1998. Equations (1) and (2) are typically applied to sinogram data in known PET image reconstruction methods. In the list-mode reconstruction algorithm, an LOR may be considered infinitely thin. Thus, the probability of detecting two pairs of coincidence events by the same LOR is negligibly small. In other words, one LOR can see either one pair of coincidence events or none at all. The value of λi, is 1 if a pair of coincidence events is detected by the ith LOR. Otherwise, the value of λi is zero. Based on this observation, the iterative update equations for list-mode MLEM and OSEM may be equations (5) and (6) respectively.
                                          F            ^                    j                      k            +            1                          =                                                            F                ^                            j              k                                                      ∑                i                            ⁢                                                          ⁢                              P                ij                                              ⁢                                    ∑                              i                ,                                                      λ                    i                                    =                  1                                                      ⁢                                                  ⁢                                          P                ij                                                              ∑                                      j                    ′                                                  ⁢                                                      P                                          ij                      ′                                                        ⁢                                                            F                      ^                                                              j                      ′                                        k                                                                                                          (        5        )                                                      F            ^                    j                      k            ,                          m              +              1                                      =                                                            F                ^                            j                              k                ,                m                                                                    ∑                                  i                  ∈                                      S                    m                                                              ⁢                                                          ⁢                              P                ij                                              ⁢                                    ∑                                                i                  ∈                                      S                    m                                                  ,                                                      λ                    i                                    =                  1                                                      ⁢                                                  ⁢                                          P                ij                                                              ∑                                      j                    ′                                                  ⁢                                                                  ⁢                                                      P                                          ij                      ′                                                        ⁢                                                            F                      ^                                                              j                      ′                                        k                                                                                                          (        6        )            These adapted iterative update equations may be applied to the list of coincidence event data to reconstruct PET images. Unlike the sinogram-based reconstruction, the adapted equation (5) uses an individual event (or photon) for each iterative step. And the adapted equation (6) uses a subset (or a group of photons) for each iterative step. However, the above formulations take into account only the true coincidence events, ignoring the scatter and random events. That is, the existing list-mode iterative image reconstruction methods do not guarantee reliable results in situations where the random and scattered coincidences form a substantial fraction of the acquired data set.
Rahmim et al [A Rahmim, M Lenox, A J Reader, C Michel, Z Burbar, T J Ruth and V Sossi, “Statistical list-mode image reconstruction for the high resolution research tomography” Physics in Medicine and Biology. 49, pp 4239-4258, 2004] have proposed using the delayed window method for random coincidences correction for conventional (NON TOF) list mode reconstruction. The method they propose, while achieving random coincidences corrections, has two drawbacks. First, the method involves the acquisition and storage of a separate list of events that approximate the random coincidences. As a result the memory storage as well as data access requirements for image generation are increased. Second, since the random coincidences estimate is computed from a noisy acquisition, it can result in increased image noise.
In view of the foregoing, it would be desirable to provide a more practical and efficient solution for TOF-PET image reconstruction.